ModeDing of turbulent flow with suspended cohesive sediment

Transcription

1 Fine Sediment Dynamics in the Marine Environment J.C. Winterwerp and C. Kranenburg (Editors) 2002 Elsevier Science B.V. All rights reserved. 155 ModeDing of turbulent flow with suspended cohesive sediment Erik A. Toorman Hydraulics Laboratory, Civil Engineering Department, Katholieke Universiteit Leuven Kasteelpark Arenberg 40, B-3001 Leuven, Belgium Traditional (cohesive) sediment transport models contain several simplifications which are no longer justifiable when sediment concentrations or stratification effects become significant. This paper gives a rather teehuical overview of various modifications to a sediment transport model with k-8 turbulence closure, used as research tool, in order to improve the physics described by the model. The attention is focussed on the modehing of sediment-turbulence interactions. KEYWORDS sediment-laden flow, turbulence modulation, buoyancy damping, saturation, drag reduction, modehing 1. INTRODUCTION The numerical modehing of suspended sediment transport in turbulent flows has been studied since the late seventies. However, the simplifications made in the early models, which are still commonly used, are only acceptable for very low concentrations and low levels of stratification. Several consequences of the sediment-turbulent interactions have never been taken into account. Within the framework of the MAST3 COSINUS project this subject has been studied in greater depth (Toorman et al., 2002). This paper focuses on the contribution of the Hydraulics Laboratory of the Katholieke Universiteit Leuven, where a generalized 2- dimensional numerical model based on the Reynolds-averaged Navier-Stokes (RANS) equations with eddy viscosity turbulence closure, the code FENST, has been developed over the past years. The primary purpose of the model is to use it as a virtual laboratory to study sediment transport processes in a verticallongitudinal cross-section of an open-channel (e.g. an estuary). The model generated data are more detailed than can be obtained experimentally and can help understand phenomena which are observed in the field. In this context it is very important to realise that the interpretation of the model results has to be done with great care in view of the limitations and shortcomings of the model (e.g. the description of the physics therein and the numerical discretisation). The model restricts itselfto the implementation ofthe mixing-lengthand the two-equation k- 8 turbulence models, which are presently used in engineering models applied to coastal and estuarine sediment transport. This will allow a relatively easy transfer of new implementation methods and improvements into these models. Higher-order turbulence models, such as

2 156 Reynolds stress models (e.g. Galland et al, 1997), do not yet perform any better and at a much higher computational cost. Details of the numerical model FENST are presented, showing the major differences with the traditional models. First the model equations are described in greater detail. Following, the modelling of buoyancy effects and the treatment of boundary conditions is discussed. Model results of steady open-channel flow are presented and discussed. Certain results, which reveal some remarkable features, are studied analytically in further detail. Their implications are shown to help understand the concepts of drag reduction and saturation in sediment-laden turbulent flows. 2. MODEL DESCRIPTION The present study has been carried out with the research code "FENST-2D" (Finite Elements for Navier-Stokes, Sediment Transport and Turbulence- 2-Dimensional), developed by the author. It solves the full hydrodynamic equations for the velocity components and the pressure, and the sediment transport equation (or sediment mass balance) for the sediment concentration. Various turbulence closures are implemented: the Prandtl mixing-length (PML) model, the standard k-6 model for high turbulent Reynolds number shear flows, various low- Reynolds number (low-ret) k-6 models and a hybrid two-layer approach, where a one-equation turbulence model coupled to a mixing-length model is applied in a layer near solid boundaries. The model is second-order accurate in space (using 9-noded Lagrangean elements) and uses first order implicit time stepping. Various numerical stabilisation techniques have been implemented (streamline-upwinding for advection-dominated flows, self-eliminating artificial diffusion, pseudo-time stepping and relaxation techniques for the k-e model). Nevertheless, the intrinsic numerical diffusion of the various schemes is very low, in particular compared to commercial CFD codes where robustness is the primary concern. Moving boundaries (e.g. tidal effects) can be dealt with using the Arbitrary Lagrange-Euler method (Huerta and Liu, 1988; Toorman, 1993) Equations The hydrodynamics of the sediment-laden flowing water is modelled using the mixture theory. As both the fluid and solid phases are assumed incompressible, the suspension mass conservation reduces to the continuity equation, despite the fact that the suspension density p may vary (Toorman, 1996). The suspension continuity reads: au~ =0 (1) Oxj The momentum conservation of the suspension is described by the Reynolds-averaged Navier- Stokes equations: = p(. + v,), ou, I! Op r,ga~ (2)

3 157 Conservation of sediment is expressed by the sediment transport equation: OC OC_ 0 I OC ) ~+ Uj ~ - (v + v, ) + w, C6jz Ot Ox j Ox j cr (3) In these equations the following notations are used: U = the mean velocity, p = pressure, t = time, xj = the components of the co-ordinate vector, v = the kinematic viscosity of the suspension, vt = the eddy viscosity, ~ = the turbulent Schmidt number (the ratio of vt to the eddy diffusivity of the sediment particles), C = concentration by mass, ws = the representative mean settling velocity, 6~y = the Kronecker delta. For cohesive sediments, the settling velocity preferably is obtained from a flocculation model. This set of equations requires a turbulence closure. In this paper the focus is on the application of the k-e turbulence closure, where the eddy viscosity is calculated as: vt = fu cu kz/6 (4) The k-6 model solves the conservation of turbulent kinetic energy k: o I --"~ UJ ~ OXj = OXj (V q" Vt k OXj J e at a -~176 (5) and its dissipation rate 06 06_ 0 / 06) 1 ---II-Uj----- (v -.~ Ot Oxj Oxj o'g~xj ~t - Vt ) dr- (flclp+c3g- f2c2oe) (6) where: Tt = k/e is the (high-reynolds) turbulence time scale, P is the shear production and G the buoyancy term, respectively defined by: ou, ouj )oui P=v t + Oxj Ox i Oxj (7) G = g vt Op per, Oz (8) with g the gravity constant and z the vertical co-ordinate. The remaining coefficients have been determined semi-empirically and are taken as c u = 0.09, Cl = 1.44, c2 = 1.92, crk = 1.0 and o-~ = 1.3, which are commonly used values for turbulent shear flows (Rodi, 1980; Chen and Jaw, 1998). The value of c3 in stable stratified shear flows is generally somewhere in the range (Rodi, 1980). Uittenbogaard et al (1992) argue that the Richardson number effect is negligible for the scales where e is important. Hence, they suggest c3 = 0, which is chosen here also. It is computationally advantageous and seems to perform satisfactorily (e.g. it yields the expected

4 158 saturation value of the Richardson number; see below). The factors fu, J] and J~ are correction functions for the low-reynolds formulation (see further); their value is 1 for the standard high- Reynolds form. The above conservation equations neglect as usual certain contributions with regard to density variation effects. For the applications under consideration (i.e. sediment-laden openchannel flow without density fronts) these additional terms are not important Buoyancy effects Despite the presence of an explicit buoyancy term in the TKE equation, empirical damping functions, i.e. Fm for momentum and Fs for mixing, such as required for the PML model, are still needed for the determination of the turbulent Schmidt number o-s = or0 Fm/Fs, (where Go = the neutral turbulent Schmidt number) which occurs in the sediment transport equation and the buoyancy term, and for the 6 near-wall boundary condition, eq.(12) (see further). Traditionally, they are determined as a function of a Richardson number, usually the gradient Richardson number Ri. Details on the determination of the damping functions can be found in (Toorman et al., 2002). However, this Richardson number may not be the most suitable one. The general flux Richardson number is defined as the ratio of buoyancy to all production mechanisms (Ivey and Imberger, 1991), i.e.: 1 Rf = ~ (9) 1-~/G and is even valid under non-equilibrium conditions. Traditionally, inertia and diffusion are neglected, resulting in the definition Rfp = -G/P. This is acceptable for slowly varying shear flows, except in open-channel flows near the free surface where diffusion dominates over shear production, because there is no shear at the surface (in the absence of wind stresses). Consequently, Richardson numbers based on P only produce too high values near the surface, resulting in excessive turbulence damping which does not allow mixing up to the surface. This can be avoided by defining a generalised gradient Richardson number, in analogy with the generalised Schmidt number, as: Ri - -cr G -~=crrf D+P (10) where D is the diffusion term in the TKE equation. Damping functions based on the generalised flux Richardson number seem to be better founded (e.g. Ellison, 1957) and more general, but need to be converted into a form where the Schmidt number can be computed explicitly. This is still under investigation (Toorman, 2000b) Free surface boundary treatment The free surface boundary conditions for the k-e model are not well established. From measurements it is known that both k and s are non-zero at the surface (Nezu and Nakagawa, 1993). The semi-empirical method for the determination of k and e at the surface, proposed by Rodi (1980), has become popular. Nevertheless, setting k and ~ zero gives comparable results and is preferred because of its simplicity. It should also be emphasised that in 2DV

5 calculations, the simulated flow is not completely realistic, because turbulence is always 3D and channel widths are finite, generating secondary currents, which are not taken into account by the present model. This will be studied in the future with a quasi-3d model Bottom boundary treatment As the standard k-6 model is only valid for high turbulent Reynolds numbers (i.e. Ret = kz/v6 > 100), numerical problems occur where this condition is not fulfilled. This is the case near a solid boundary, where a viscous boundary layer is formed as the velocity at the boundary itself is zero. Therefore, the wall layer is skipped by the k-s turbulence model and boundary conditions are imposed far enough from the wall where the turbulence is assumed to be fully developed. From shear flow experiments it is known that this distance can be estimated as 6 = 60 flu, (with u, the shear velocity - (to~p) 1/2 and r0 the bottom shear stress). The boundary conditions are obtained from assuming local equilibrium between production, buoyancy destruction and dissipation of TKE. Toorman (1999) shows that diffusion remains negligibly small until saturation (see section 3.2 for its characterisation). Hence, the boundary condition for zthen is: 159 L(OU)2(1-Rf) 6=P+G=P(1-Rf)= ~,Oz ) (11) The eddy viscosity and velocity gradient are estimated using the consistent PML approximation (Toorman et al., 2002). Hence, the boundary condition becomes: u. ~ e = (1 - Rf) (12) F, xzh where Fm = the momentum damping function, tc = the von Karman constant (= 0.41) and Zb = the level of the near-bottom boundary node. The boundary condition for k is obtained from combining the previous expression with the definition of the eddy viscosity, eq.(4):,2 These boundary conditions require the knowledge of the shear velocity. Traditionally, u, is obtained from the wall shear stress, approximated by the PML model, where the velocity gradient is obtained by elimination of u, between the log-velocity profile and its derivative. This method works adequately as long as stratification effects are negligible. When buoyancy damping at the near-wall node is significant, unacceptable errors are made and the shear velocity is overestimated (figure 1). Usually, the layer between the fully-developed turbulent layer and the wall, which contains the viscous sublayer and the transition layer, is not solved. Because the highest sediment concentration usually is found at the bed, this wall layer may contain a non-negligible amount of the total sediment load in suspension. Therefore, this layer should be included into the computational domain. In the present model a single-element layer is assigned to the wall layer.

6 160 In this layer the eddy viscosity is computed with the PML model. Boundary conditions for the velocity field are then imposed both at the bottom (non-slip condition, i.e. zero velocities) and at the interface between the wall layer and the turbulent layer. This is a cheap alternative for using low-reynolds near-wall turbulence models which require too much grid refinement and which are not adapted for sediment-laden flow (see section 4.2). The boundary condition for the velocity (tangential to the bottom slope) at the interface is obtained using the consistent law-of-the-wall profile: U= u-z-" ln( Z-~ I,,: L~o) (14) where z0 = the characteristic roughness height of the bottom and ot = the friction correction factor, which accounts for the buoyancy effects and can be related to the momentum damping function, o~ can be expressed by an empirical relationship, derived from numerical experiments, which is a function of the Richardson number and the ratio wju, (Toorman, 2000; Toorman et al., 2002). Equation (14) remains applicable up to saturation. Another source of error is the mesh size near the wall. A sensitivity analysis shows that with increasing thickness of the computational wall layer, compared to the actual thickness of the real sublayer, the error in the estimation of u, increases considerably. Besides the fact that the estimation of the velocity gradient is less accurate, another source of error is the traditional neglection of the pressure gradient contribution. The estimation of u, can be improved considerably by solving the local stress balance, i.e.: I/ U, = V + C u 6 az p~x (15) The corresponding error on u, remains < 2% for z+ up to 1000 (for unstratified conditions; for stratified conditions < 10% in the extreme case of saturation). Sediment exchange with the bed is possible by equalling the bottom flux to the net deposition/erosion flux (a Neumann type boundary condition). The deposition flux is assumed to equal the settling flux wsc at the bottom, while the erosion flux is modelled with a traditional empirical erosion law extended with a contribution of the deposited sediment which remains mobile. 3. FULLY-DEVELOPED OPEN-CHANNEL FLOW Steady-state calculations of sediment-laden turbulent open-channel flow have been performed. The water depth h = 16 m. A sensitivity analysis has been carried out for the parameters shear velocity (u,), settling velocity (Ws) and sediment load (or depth-averaged concentration). Various observations have been made Drag reduction The effect of the implementation of the consistent bottom boundary treatment is significant, as is illustrated by figure 1. The inconsistent method overpredicts the shear velocity

7 increasingly with increasing stratification in the near-wall node up to 70% at the saturation state. This implies that the bed shear stress in the latter case is three times overestimated, which would have enormous implications for the correct prediction of erosion and deposition. Notice furthermore the change (i.e. steepening) of the slope of the profiles, which is equivalent to the decrease of the von Karman coefficient from tcto Fmtc (Toorman, 1999). The reduction of the shear velocity for the same energy input is known as drag reduction. Various conditions are now known which can cause this phenomenon. Drag reduction in shear flows of clay suspensions has been reported in the literature (e.g. Gust, 1976; Wang et al., 1998; Li and Gust, 2000). Its practical importance is illustrated by the fact that the friction coefficient for some sediment-laden rivers in China had to be modified to those for bottoms smoother than glass (Wang and Larsen, 1994). The exact mechanism for drag reduction is still not understood. Gust (1976) tried to explain drag reduction in terms of energy dissipation within the flocs, in analogy with polymers. This, however, seems unlikely considering their weak strength. Best and Leader (1993) proposed that the increased spacing between low-speed streaks and decreasing bursting rate in the inner wall layer, observed in polymer flow studies, could also occ~ in clay suspensions. Li and Gust (2000) conclude that there is no experimental evidence to support any of the hypothesized mechanisms for drag reduction in clay suspension flow. Model results with the inconsistent treatment produce drag reduction one order of magnitude smaller than observed. Therefore, it was believed up till now that buoyancy could not account for the drag reduction. However, the present results yield the right order of magnitude and even correspond quantitatively very good with the recent data of Li and Gust (2000), who also found a maximal drag reduction of 70% i<---level (Zb = 0.08 m) where boundary conditions are imposed xe turbulent flow =;mp=e-d A E traditiona! (inconsistent) --- consistem without friction correction m consistem + friction correction. l I I I 10 z (m) 100 Figure 1. Computed velocity profiles (on semi-log scale) of fullydeveloped open-channel flow driven by a constant horizontal pressure gradient (dp/dx = 0.04 Pa/m, h = 16 m) for four different sediment loads (mean concentration from bottom to top = 19, 26, 68 and 118 mg/1).

8 162 A systematic study of steady state calculations with stepwise decreasing shear velocities clearly shows how the depth averaged velocity decreases down to a minimum and then increases again due to the dominance of the drag reduction. Another reason for the thickening of the sublayer is the increase of the suspension viscosity due to the high concentrations near the bottom, which is calculated according to the Krieger- Dougherty model (Krieger, 1972): v = v (1 - r / a~ lmax ]-25~.. / w (16) where Vw = the kinematic viscosity of water and ~ = the solids volume fraction. The maximum volume fraction g~max is assumed to be 0.74, corresponding to the maximum packing fraction of spheres. For cohesive sediments this relationship needs to be modified, because the suspended particles are floes, containing immobilised pore water. Hence, ~b needs to be replaced by the volume fraction of the floes, q~nax then corresponds to the gelling volume concentration (again of the floes), but may need to multiplied by a factor >1 to account for the fact that the spacefilling structure itself is not rigid Saturation By reducing uo or by increasing ws or the depth-averaged concentration C, while keeping the other parameters constant, qualitatively similar results are obtained, i.e. an increase of the stratification with subsequent increasing buoyancy damping of the turbulent kinetic energy. This cannot be continued endlessly. The total amount of available TKE can only keep a limited amount of sediment in suspension. This limit is known as the saturation or capacity limit; any amount in excess will deposit (Cellino and Graf, 1999). This is illustrated by figure 2, which shows the computed steady state concentration profiles at the same energy input (constant u.) for various sediment loads. For small loads the concentration is relatively homogeneous. With increasing load, the profile is more and more approaching a limiting profile, starting near the surface. A remarkable feature is the fact that the corresponding Richardson number tends to homogenise over the entire water column, except near the free surface, when approaching the saturation limit (figure 3). This has been investigated analytically by looking at the implications of the condition drfldz = 0, which can be rewritten as dg/dz = -Rf dp/dz. After substitution of (7) and (8) and making a few simplifying assumptions (i.e. ws and ~c are constant, p/pw ~ 1 and du/dz = u./~), this leads to a parabolic eddy viscosity distribution, given by (Toorman, 1999): v, =o'sw, z(1-z/h ) (17) This is like the PML eddy viscosity distribution in which the von Karman constant x has been replaced by ~Ws/U,. The corresponding concentration profile can be computed by integration of the equilibrium sediment flux, where (17) is used as eddy viscosity distribution. It is found to be a Rouse profile with Rouse parameter Z = ~WshCU, = 1, which is the fully saturated limiting profile (figure 2). Attempts to increase the sediment load beyond fail because the turbulence damping at the bottom becomes too strong due to the increasing sediment load which cannot be entrained into the turbulent flow.

9 ~ -to -r E E E E+O0 CONCENTRATION (g/i) 1.00E+01 Figure 2. Computed concentration profiles for various sediment loads corresponding to a depth-averaged concentration (from left to fight) of 4.88, 16.3, 48.8, 163, 326 and 488 mg/1 in fully-developed turbulent sediment-laden openchannel flow with u, = 0.01 m/s and ws = 0.5 mm/s, using Munk-Anderson damping functions. Dashed line = limiting Rouse profile (Z = I) ~" Rf ~,1o ~ Rf=-G G/(9 /(D + P) 2 Rf = ~/( ]-e /G) o FLUX RICHARDSON NUMBER 0.8 Figure 3. Generalised flux Richardson number Rf and corresponding Rfp for the same conditions as in figure 2. Increasing sediment load from left to fight.

10 164 The physical meaning of the value of Rffor which this limit occurs is still unclear. The value found for the example is Rfs = 0.245, which is close to the magical value of 0.25 for which total collapse of turbulent mixing is claimed by some. There seems to be some dependence on certain model parameters (e.g. for c3 = 1, one obtains Rf~ ~ 0.5). The model results indicate that stable turbulent flow can exist near the surface where diffusion dominates for Rf> Rf~ and > Therefore, Rf~ cannot be the critical value for total turbulence collapse. That critical value seems to be simply Rfc = 1. It is therefore suggested that the results can only be understood in terms of the vertical gradient of Rj5 a stable situation seems only possible when arflcgz > O. The condition arf/cgz = 0 should be related to the minimal energy required to keep a certain amount of particles in suspension. The range 0.25 < Rf< 1 probably corresponds to the "Richardson number hysteresis" phenomenon, described by Woods (1969). It is hypothesised that Rf~ is only an apparent critical number, because the model results indicate the existence of a thin near-wall layer over which Rfrapidly reaches the physically expected critical value 1. A further discussion can be found in (Toorman, 2000a). The slope of the corresponding velocity profile on a logarithmic z-scale has also increased from 1/x to u,/~ws. This is new theoretical evidence that the von Karman coefficient indeed decreases with increasing sediment load. All these theoretical findings are reproduced by the model (Toorman, 1999). From the sediment transport equation one can determine the local equilibrium between settling and turbulent mixing. Teisson et al. (1992) call the corresponding concentration for the critical value of Rf where "turbulence collapse occurs" the (local) saturation concentration. As argued above, the interpretation of the critical value of Rfshould be relaxed, because turbulence may persist for higher values of Rf. Therefore, saturation is physically related to optimal energy conditions instead of turbulence collapse. Since Rf is constant at the saturation limit, one can integrate this relationship over the water depth, giving the total maximum suspended sediment load per unit area: C~---h= P'u~'U Rye, wsgap s / Ps (18) m where U is the depth-averaged velocity. With exception of the factor Rfsat, this is identical to the auto-suspension criterion for density currents, as given by Parker et al. (1986). It should be realised that the described numerical experiment is not very representative for most field conditions. Steady flow conditions do not occur in coastal and estuarine areas, where the flow is subjected to tidal variations, i.e. periodic acceleration and deceleration. Therefore, concentrations in the field may be higher or lower than the steady state values depending on the flow conditions and the time scales for entrainment and settling relative to the tidal period. Even when saturation would have occurred at steady state, higher concentrations than the corresponding saturation value may be found in the field. 4. LOW-REYNOLDS EFFECTS 4.1. Modelling the wall layer In the previous model description the wall layer was treated on a single-element layer grid. The parabolic interpolation function over this layer cannot follow the exact profiles of any of

11 the variables. As long as the thickness of the sublayer is much smaller than the wall layer grid size, the error remains acceptably small. Problems occur when the sublayer thickness grows due to the reduction of the shear velocity and its dimension becomes of the same order of magnitude as the grid size. In physical practice, the problem of the thickening of the sublayer may occur during flow reversal around slack water. In numerical practice, problems may be encountered when approaching saturation due to the large gradients near the bottom. The only way to improve the model's performance is to solve the wall layer more correctly on a refined grid in order to approximate the profiles better. A first approach, which has been investigated, is the low-reynolds k-e model, where the damping functions fu, fl andj~ are introduced to correct the original k-e equations, as shown in equations (4) and (6). Various forms have been proposed in the literature, most of which contain correction factors which are a function of the distance from the wall. These have been evaluated against Direct Numerical Simulation (DNS) data (Toorman, 2000c), which resulted in finding a relationship between J~ and a new formulation of the realisable time scale, which allows transition from the fully-developed turbulent time scale k/e to the Kolmogorov time scale (v/e)1/2: 165 r. f T _ I vt / cite f lt -].- v (19) where fr = A "1 -" (1 + 1/c u Ret) 1/2 the realisable time scale factor, with Ret = k2/ve the turbulent Reynolds number. The analysis also revealed that the model parameters o-k and o-~ cannot be constant, because the assumption of isotropy of turbulence is no longer correct in the wall layer. It would not be surprising that these model constants need to be modified as well, i.e. made a function of Rf, in the turbulent layer when highly stratified. Because of the high degree of refinement in the wall layer, this method is not suitable for large-scale applications. Another method which has been investigated is the two-layer approach, in which the wall layer is solved with a modified mixing-length model in replacement of the 6 equation (e.g. Chen and Patel, 1988; Rodi, 1991). The eddy viscosity in the wall layer is computed as: v, =~,ffe, (20) and the dissipation rate as: 6 = k 3'2 / g, (21) The models in the literature use a length scale of the form: g~ = Kc-9/4zO - exp(-r / A )) (22) where Rz = kl/2z/v = a turbulent Reynolds number. For the length scale g, two different forms have been proposed, either like the one for ~r or:

12 166 ICC -3 / 4 Z g, = ~ (23) I+A~/R~ The values of the model parameters A u and A, chosen by various researchers differ. The wall model and the full k-6 model are matched at a distance where Rz > 200 (z ~ 135) or where vt/v > 30. The model still requires about 10 grid points in the wall layer. Further details and references can be found in Section 4.4 of (Chen and Jaw, 1998). The model has been tested successfully for fully-developed turbulent shear flow (Toorman, 1998). The major problem with both methods is the limitation of their validity to homogeneous fluids for which the damping functions have been calibrated. Effects of buoyancy are not included and require suitable data, in particular within the wall layer, which, as far as known, are not available. Hopefully, future DNS simulations of particle-laden flows at realistic Reynolds numbers will provide useful data. Based on these data damping functions can be proposed in analogy with those for the Prandtl mixing length turbulence closure. The other concern is the required grid refinement. More work needs to be done in order to find ways to minimise the number of grid points in the wall layer without loss of accuracy Modelling lutoclines Model results for unsteady open-channel flow simulations indicate that laminarisation may also occur around lutoclines when the density gradient is too large. Results obtained with the present model are shown in (Toorman et al., 2002). Low-Ret formulations suitable for these conditions are not yet available. Therefore, a simple, heuristic method has been implemented consisting of the introduction of local artificial diffusion which is inverse proportional to the local value of c. This added diffusion only plays a role where turbulence is excessively damped and stabilises the solution of the k-6 model. The occurrence of the numerical problem and its cure also depends on the grid size at the location of the lutocline. Internal wave corrections may be necessary and helps reducing the problem. This is discussed in more detail in (Toorman et al., 2002). 5. CONCLUSIONS The consistent modelling of turbulent flows with suspended sediment particles, using the k-e turbulence model, requires several modifications to the standard implementation. The bed boundary conditions are adapted and the computation of the shear velocity is corrected following the consistent implementation of buoyancy damping functions. Previously, it was thought that several features, such as the decrease of the von Karman coefficient and drag reduction, could not be explained by buoyancy effects alone. This conclusion was based on application of the traditional, inconsistent modelling approach. Numerical experiments with the consistent implementation shows that this is not true, i.e. buoyancy effects are very important, even at very low concentrations. Other turbulence modulation mechanisms, which occur also in homogeneous suspensions, are less important in the present context because of the low concentrations. From the analysis of experimental data of Cellino (1998), it is concluded that these effects are no longer negligible in the oversaturated bottom layer (Toorman, 2000a). Another correction, based on the local stress balance, is carried out allowing a more accurate estimation of the shear velocity on a coarse grid. The wall layer is explicitly solved using the

13 Prandtl mixing length concepts in order to include the sediment in this layer. The Richardson number has been generalised in order to account for the dominance of diffusion of turbulent kinetic energy at the free surface. All these modifications can easily be implemented into presently used engineering models which use the k-6 turbulence closure. Many of them can even be implemented for models using the PML model (Toorman et al, 2002). Coastal and estuarine applications generally need to be solve in three dimensions and require a large number of grid points, which results in a relatively coarse vertical discretisation. Its implications in view of the presently proposed improvements are discussed in (Toorman et al, 2002). Numerical simulations of steady-state conditions reveal that saturation can be obtained and seems to be charactefised by the condition of a zero vertical gradient of the Richardson number. The theoretical analysis of this condition shows that the yon Karman parameter indeed decreases with increasing sediment load. The data also suggest that turbulent flow can be maintained for flux Richardson numbers between the saturation value and the maximum value of 1 where buoyancy destroys shear production, as long as the vertical gradient of Rf remains positive. Other steady-state computations confirm that drag reduction takes place as a result of the buoyancy damping and its magnitude is found to be of the same order as observed in laboratory and field conditions. This contradicts the previous hypothesis that buoyancy damping cannot account for the high degree of drag reduction measured. The model starts having numerical problems when saturation is approached or when the shear velocity becomes small, because the wall-layer is no longer solved properly. This topic needs further study. 167 Acknowledgements: This work has been carried out within the framework of the MAST3 project "COSINUS", partly funded by the European Commission, Directorate General XII for Science, Research and Development under contract no. MAS3-CT The author's post-doctoral position is financed by the Flemish Fund for Scientific Research. REFERENCES Best, J.L. and Leeder, M.R., 1993, Drag reduction in turbulent muddy seawater flows and some sedimentary consequences, Sedimentology, 40, Chen, C.J. and Jaw, S.Y., 1998, Fundamentals of Turbulence Modelling, Taylor and Francis, Washington D.C. Chen, H.C. and Patel, V.C., 1988, Near-wall turbulence models for complex flows including separation, AIAA J., 26(6), Cellino, M., 1998, Experimental study of suspension flow in open-channels, PhD thesis, D6pt. de G6nie Civil, Ecole Polytechnique F6d6rale de Lausanne. Cellino, M. and Graf, W.H., 1999, Sediment-laden flow in open channels under noncapacity and capacity conditions, ASCE J. Hydr. Eng., 125(5), Ellison, T.H., 1957, Turbulent transport of heat and momentum from an infinite rough plane, J. Fluid Mechanics, 2,

14 168 Galland, J.C., Laurence, D. and Teisson, C., 1997, Simulating turbulent vertical exchange of mud with a Reynolds stress model, In: Cohesive Sediments, N. Burt, R. Parker and J. Wats, eds., J. Wiley and Sons, Chichester, Gust, G., 1976, Observations on turbulent drag reduction in a dilute suspension of clay particles, J. Fluid Mech., 75(1), Huerta, A. and Liu, W.K., 1988, Viscous flow with large free surface motion, Comp. Meth. Appl. Mech. and Engrg., 69, Ivey, G.N. and Imberger, J., 1991, On the nature of turbulence in a stratified fluid. Part I: The energetics of mixing, J. Physical Oceanography, 21, Krieger, I.M., 1972, Rheology of monodisperse latices, Advanc. Colloid Interface Sci., 3, Li, M.Z. and Gust, G., 2000, Boundary layer dynamics and drag reduction in flows of high cohesive sediment suspensions, Sedimentology, 47, Nezu, I. and Nakagawa, H., 1993, Turbulence in Open-Channel Flows, IAHR Monograph Series, A.A. Balkema, Rotterdam. Parker, G., Fukushima, Y. and Pantin, H.M., 1986, Self-accelerating turbidity currents, J. Fluid Mechanics, 171, I. Rodi, W., 1980, Turbulence models and their application in hydraulics, State-of-the-art Paper, IAHR, Deltt. Rodi, W., 1991, Experience with two-layer models combining the k-6 model with a oneequation model near the wall, A/AA 29 th Aerospace Sciences Meeting, Paper AIAA Teisson, C., Simonin, O., Galland, J-C. and Laurence, D., 1992, Turbulence modelling and mud sedimentation: a Reynolds stress model and a two-phase flow model, Proc. 23rd Int. Conf. Coastal Eng., ASCE, New York, 3, ,. Toorman, E.A., 1993, Simulation of free surface flow of mud, Proc. 8th lnt. Conf. on Finite Element Methods in Flow Problems, Morgan et al. eds., CIMNE/Pineridge Press, Barcelona/Swansea, Toorman, E.A., 1996, Sedimentation and self-weight consolidation: general unifying theory, Gbotechnique 46(1), Toorman, E.A., 1998, A study of modelling (near-wall) turbulence damping, COSINUS Annual General Meeting Grenoble 1998, Book of Abstracts, pp Toorman, E.A., 2000a, Suspension capacity of uniform shear flows, Report HYD/ET/ COS1NUS4, Hydraulics Laboratory, Katholieke Universiteit Leuven. Toorman, E.A., 2000b, Parametrisation of turbulence damping by suspended sediment, Report HYD/ET/COSINUS3, Hydraulics Laboratory, Katholieke Universiteit Leuven. Toorman, E.A., 2000c, Analysis of near-wall turbulence modelling with k-s models, Report HYD/ET/COSINUS2, Hydraulics Laboratory, Katholieke Universiteit Leuven. Toorman, E.A., Bruens, A.W., Kranenburg, C. and Winterwerp, J.C., 2002, Interaction of suspended cohesive sediment and turbulence, Proc. 1NTERCOH-2000, J.C. Winterwerp and C. Kranenburg eds., Elsevier, this volume. Uittenbogaard, R.E., van Kester, J.A.Th. and Stelling, G., 1992, Implementation of three turbulence models in TRISULA for rectangular horizontal grids, Report Z162, Delft Hydraulics. Wang, Z.Y. and Larsen, P., 1994, Turbulent structure of water and clay suspensions with bed load, J. Hydraulic Eng., 120(5), , ASCE. Wang, Z.Y., Larsen, P., Nestmann, F. and Dittrich, A., 1998, Resistance and drag reduction of flows of clay suspensions, J. Hydr. Eng., 124(1),

15 Woods, J.D., 1969, On Richardson's number as a criterion for laminar-turbulent-laminar transition in the ocean and atmosphere, Radio Science, 4(12),